On minimal Ramsey graphs and Ramsey equivalence in multiple colours

Abstract

Abstract For an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences. For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H. For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number. The collection $\{\mathcal M_q(H) \colon H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where $\mathcal M_q(H)$ denotes the set of all graphs that are q-Ramsey-minimal for H. We also address the question of which pairs of graphs satisfy $\mathcal M_q(H_1)=\mathcal M_q(H_2)$ , in which case H 1 and H 2 are called q-equivalent. We show that two graphs H 1 and H 2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.